The Zelevinsky classification of unramified representations of the metaplectic group (Q258232)

Let us denote by \(F\) a nonarchimedean local field of characteristic different than two and let \(\widetilde{\mathrm{Sp}(n,F)}\) stand for a rank \(n\) metaplectic group over \(F\), i.e., a unique non-trivial two-fold central extension of rank \(n\) symplectic group over \(F\).NEWLINENEWLINEAn irreducible genuine representation \(\pi\) of \(\widetilde{\mathrm{Sp}(n,F)}\) is called unramified or spherical if it has a nonzero vector fixed under a maximal compact subgroup of \(\widetilde{\mathrm{Sp}(n,F)}\). In the paper under review, the authors obtain a classification of the unramified representations of the metaplectic group analogous to the one obtained by \textit{G. Muić} in the classical group case [Trans. Am. Math. Soc. 358, No. 10, 4653--4687 (2006; Zbl 1102.22014)].NEWLINENEWLINEAccording to the obtained classification, which is mostly based on the Jacquet modules method, every unramified representation can be obtained as a fully induced representation from unramified characters of general linear groups and a negative unramified representation of metaplectic groups of a smaller rank. An explicit description of negative unramified representations is also provided.